Question

How to evaluate lim(x -> [infinity]) of (x⁴ * x³)/(12x³ + 128)?

a) [infinity]
b) 0
c) 12
d) 1

Answer 1

The limit as x approaches infinity of (x^4 * x^3)/(12x^3 + 128) is evaluated by dividing both numerator and denominator by x^3, simplifying the expression, and noting that as x approaches infinity, the limit equals infinity. Therefore, the correct answer is option a) infinity.

Step-by-step explanation:

How to Evaluate a Limit as x Approaches Infinity

When evaluating the limit lim(x -> ∞) of (x⁴ * x³)/(12x³ + 128), we are dealing with a limit where x approaches infinity. To solve this, we can apply the concept of polynomial division. Since the highest power in both the numerator and denominator is x³, we can simplify the equation by dividing each term in the numerator and the denominator by x³.

When we do this division, the function simplifies to:

lim(x -> ∞) of x⁴ / x³ = x
and
lim(x -> ∞) of x³ / x³ = 1

So the original expression simplifies to:

lim(x -> ∞) of (x * 1)/(12 + 128/x³)

As x approaches infinity, the term 128/x³ approaches 0 since any constant divided by an infinitely large number becomes negligible.

Therefore, our limit simplifies further to:

lim(x -> ∞) of x/12

Since x is approaching infinity, we can see that this limit will go to infinity as well. Therefore, the correct answer is a) ∞.